
The secure domination number of \(G\)  is defined to be \(\gamma _s(G)=\gamma _{(1,0)}^s(G)\) . In this case, for any secure \((1,0)\) -dominating function \(f(V_0,V_1)\) , the set \(V_1\)  is known as a secure dominating set. This concept was introduced by Cockayne et al. in [1]} and studied further in several papers, including among others, [2]}, [3]}, [4]}, [5]}, [6]}, [7]}.

The secure total domination number of a graph \(G\)  of minimum degree at least one is defined to be \(\gamma _{st}(G)=\gamma _{(1,1)}^s(G)\) . In this case, for any secure \((1,1)\) -dominating function \(f(V_0,V_1)\) , the set \(V_1\)  is known as a secure total dominating set of \(G\) .
This concept was introduced by Benecke et al. in [8]} and studied further in several papers, including among others, [9]}, [10]}, [11]}, [6]}, [13]}.

The weak Roman domination number of a graph \(G\)  is defined to be \(\gamma _{r}(G)=\gamma _{(1,0,0)}^s(G)\) .
This concept was introduced by Henning and Hedetniemi [14]} and studied further in several papers, including among others, [15]}, [4]}, [5]}, [18]}.

The total weak Roman domination number of a graph \(G\)  of minimum degree at least one is defined to be \(\gamma _{tr}(G)=\gamma _{(1,1,1)}^s(G)\) . This concept was introduced by Cabrera et al. in [10]} and studied further in [20]}.

The secure Italian domination number of \(G\)  is defined to be \(\gamma _{_I}^s(G)=\gamma _{(2,0,0)}^s(G)\) . This parameter was introduced by Dettlaff et al. in [21]}.

